**I**n this post we are going to discuss about the **inequalities** in absolute value of real numbers which we come across while solving the questions related to **inequalities.**

** Absolute value** of real number \(\;x\;\) also known as **numerical value or modulus of real number** is denoted by \(\;|x|\;\) is defined as

$$|x|=\begin{cases}x \;&\;\text{if }\;x\geq 0\\-x;& \;\text{if}\;x\lt 0\end{cases}$$

in a fun way we can say the \(\;|x|\;\) is a person with positivity, which take negativity and convert into positivity, for example \(\;|-2| = 2 \;\), and \(\; |2| = 2\;\) , so this function only act on negative values , but keeps positive values as it is.

After this we are going to discuss some inequalities properties used in absolute value for real numbers .( Here we are specific about **real numbers **because for **complex numbers** these properties may vary)

if \(\; a, b \in \mathbb{R}\;\) then

1) \(\displaystyle |a|^{2} = |-a|^2=a^2\)

2) \(\displaystyle |ab|=|a||b|\)

3) \(\displaystyle \left |\frac{a}{b} \right |= \frac{|a|}{|b|}\) provided \(\; b \neq 0\)

**also known as Triangle Inequality**

**also known as Triangle Inequality**

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